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APPLIED HYBRID METHODS TO SOLVING INITIAL-VALUE PROBLEM TO INTEGRO-DIFFERENTIAL EQUATIONS OF VOLTERRA TYPE

APPLIED HYBRID METHODS TO SOLVING INITIAL-VALUE PROBLEM TO INTEGRO-DIFFERENTIAL EQUATIONS OF VOLTERRA TYPE
Galina Mehdiyeva, доктор, профессор

Imanova Mehriban, кандидат физико-математических наук

Ibrahimov Vagif, доктор физико-математических наук, профессор

Бакинский государственный университет, Азербайджан

Участник конференции

UDC 519.62/.642

During consideration of the chronology of scientific works we may note that scientists have studied integro-differential equations as more comprehensive than differential and integral equations. That is why differential equations are more often at the basis of studies, then integral and integro-differential equations. As the result, the majority of specialists substitute the solving of integral and integro-differential equations by solving differential equations. To widen this subject authors have presented the general hybrid method for solution of integro-differential equations and determination of their equality with to the above-mentioned equations. The certain stable hybrid method considered in the article allows to use the point of linkage in the lower limit with split-hair accuracy. From the other side at k=1 (with the use of the linkage point) one-step method is used, when p=6.

Keywords: nonlinear integro-differential equations, hybrid method, ordinary differential equations, multistep methods, relation between order and degree for hybrid methods.

 

Consider to solving of the following Volterra integro-differential equation of first order:

                             (1)

Suppose that equation (1) has a unique solution defined on the segment [x0,X] and satisfying the following initial condition:

y(x0)=y0                                                                             (2)

To determine the numerical solution of the problem (1) - (2) assume that the continuous on totality of arguments, functions ƒ(x.y) and  K(x,s,y) defined in the domains

respectively, and also have continuous partial derivatives to up some order p+1, inclusively. The segment [x0,X] with a constant step-size h>0 is divided into N equal parts and mesh points define in the form:xi=x0+ih (i=0,1,...N).

To calculate the approximate values ??of the solution problem (1) - (2) used certain  formulas for, which are denoted by y an approximate, but through y(xi) the exact value of the solution of problem (1) - (2) at the mesh points xi=x0+ih (i=0,1,...N).
Beginning with V.Volterra, s work (see[1]), published in 1887 to present time, scientists engage for the investigation of approximate solutions of problem (1) - (2), constructed methods for the solving of equation (1) (see, for example[2]-[7].). But construct an effective method satisfying certain requirements is one of the based questions of modern computational mathematics. Therefore, scientists are often turning to the construction of numerical methods for the solving of problem (1) - (2), which has some additional properties. One of such methods is the hybrid methods, which applying to solve the problem (1) - (2) offer by Makroglou and developed in the works [8]-[9]. Here be in progress these researches constructed of stable hybrid methods with high accuracy and also constructed the specific methods with a certain accuracy, which are illustrated on the model problems.

In the case ?=0 the equation (1) is converted to a differential equation to the study, which engaged many well-known scientists: N.Tusi, I.Newton, A.C.Clairaut, G.W.Leibniz, L.Euler, J.L.Lagrange, A.L.Cauchy, J.C.Adams, C.Runge, W.Kutta, etc. For the investigation of numerical solution of ordinary differential equations they was construct numerous methods with different properties. Therefore, for solving integral and integro-differential equations are often used numerical methods of differential equations. This approach is explained with the present of ordinary differential equation by the  ??integral equation of the next from:

                                      (3)

which is obtained from the differential equation by integrating  on the segment [x0,X]. If equation (3) rewrite in a more general form:

                                               (4)

then we can receive equation of type (1) from it by differentiation. Given these connections between the equations (1), (3) and (4), here to consider application of the following hybrid method

                                       (5)

to the solving of the problem  (1) - (2).  Note that the method (5) is applied to the  solving of initial value problem for ordinary differential equations of first and second order (see [10]-[12]), and also  to the solving of equation (4). The hybrid method  used  by Makroglou  to  solve  the problem  (1) - (2), may  be  received from the  method  (5) in particularly for

1. Application of hybrid methods to solving Volterra integro-differential equations.
As is known, one of the first numerical methods for solving equation (1) is constructed and investigated by V.Volterra. For this purpose, Volterra used the method of quadratures, is consisted in a replacement an integral by some integral sum, which in one variant has the following form:

                                   (6)

here the quantities ai=(i=0,1,2...N) are the real numbers, but Rn-is the  remainder term. If to the solving of the problem (1) - (2) apply k-step method with constant coefficients and taking into account the method of quadratures defined by the formula (6), then we have:

                                                       (7)

Here,  are the coefficients  make up coefficients of the quadrature formula, and coefficients of k-step method, but ?i?i(i=0,1...k) the coefficients of the k-step method. It is easy to remark that the while crossing from the current mesh point to the next amount of computational work is increased, since the second sum in method (7) depends on the variable n. For the relieve from these lack in [9]for solving of the equation (4) proposed the following method:

                                                   (8)

Note that depending on the properties of the integral kernel some of the coefficients ,  will be equal to zero. If suppose that, the kernel of the integral is defined in the ?-expansion of domains G then the method  (8) can be applied to the solving of equation (4). Otherwise, the coefficients  must satisfy the condition. Note that for using of the method (8) must be known quantities y0,y1,...yk-1.By the method (8) one can calculate the values ??of variables yn+k. It is known that usually for solving problem (1) - (2) uses stable methods, but among the stable multistep methods the implicit methods are the most popular. However, when using them are meet with finding solutions of nonlinear equations, which is not always succeed. Usually in such cases, experts use  iterative methods, or methods of predictior-correctior. It is easy to show that the predictior-correctior methods in particular, may also receive from the iterative methods. But to relieve of these shortcomings of implicit methods here is proposed to use the explicit hybrid methods. Therefore, we consider hybrid methods, and their applications to the solving of the integro-differential equations. Hybrid methods can be constructed by different ways. In the work [9] consider the following hybrid method with constant coefficients:

                           (9)

For the vi=0 (i=0,1,2....k) from the formula (9) follows a well-known multistep methods with constant coefficients. Here we consider the case when there is . Usually, in the concrete methods with the maximum degree the quantity vk satisfies the condition -1<vk<0. However, in this case we obtain explicit hybrid methods. For example, from the method (9) receive the next hybrid method with the maximum degree for k=1:

                          (10)

As the remark above explicit method (10) is obtained from equation (9) for k=1 and have order accuracy p=4. This method is unique in a class methods which has the degree of accuracy p=4. For the construction of hybrid implicit methods, consider the following generalization of the method (9):

 (11)

Obviously, if ?k, and  -1<vk<0 thenthe method (11) is implicit. Appling implicit methods to solve scientific and engineering  problems has some difficulties. Therefore, usually for the construction of concrete methods considered the case ?k=0. Now consider the applications of the method (11) to the solving of problem (1) - (2). To this end, consider the following difference:

        (12)

here .

It is obvious that from the equality  (4), one can be write the following:

 (13)

Here we replace x by the ?n i.e x=?n. Then we have:

If taking into account obtained in (12), then receive the following:

              (14)

As is well known to the calculation of the integral one can apply different quadrature formulas as a method of the rectangle formulas and a trapezoid method. However, the method of quadratures can be defined as a linear combination of these methods. Then, generalizing the proposed scheme, we can write:

      (15)

If we take vi=1/2 (i=0,1,2...n) then after choosing the suitable coefficients from (15) we obtain a linear combination of generalized methods of the rectangle formulas and trapezoids (see, for example, [13, p184-186]. Similar schemes for the solving of ordinary differential equations are used by many authors (see, for example [14],[15]). Replacing the derivatives of functions by its values ??at different mesh points, applying interpolation polynomials Lagrange to calculation quantity . By using them and formula (15) in equality (14) one obtains the following formula:

    (16)

The coefficients  are some real numbers, but .
Consider the determination of the coefficients. For this purpose we consider a special case and put K(x,s,y)≡z(s,y). Then from (16) we have:

                                (17)

where


                    (18)

From equations (4) we have:

y'-g'=z(x,y)

As is follows from here the method (17) coincides with the method (5).

To determine the coefficients  of the method (17) using the method of undetermined coefficients and in the result receive the following nonlinear system of algebraic equations (see, for example [10]):

       (19)

In this system number of equations is equal p+1, and the number of unknowns is equal 4k+4. The quantity k is known, so determining the values ??of the quantity p used the values ??of the quantities k. One can show that between the quantities k and p has the following relation:

p≤4k+2                                                                         (20)

For the application of the method (16) to the solving of problem (1) - (2), the problem (1)-(2) rewrite in the following form:

Then the method (16) is apply to the solving of equation (22), and to solving problem (21) we apply the method (5) and choose the coefficients so that the coefficients in these methods coincides by  the taking into account  conditions (18).Note that if the method (5) is converges, then its coefficients satisfies the following conditions:

A: The coefficients  are some real numbers, moreover,  .

B: Characteristic polynomials

have no common multipliers different from the constant.

C: 

Consider the construction of specific methods and put k=2. Then for the determining of the coefficients we obtain the following system of nonlinear equations:

                                          (23)

By  solving these system, we find the values ??of the coefficients of the method (5), and the coefficients of the method of the type (16) determined from the system (18). Consequently, if  the method of the type (16) has the maximum degree and it,s coefficients are defined by the solution  of the system (18), then it will not be unique with the maximum degree, because the system  (18) has the solution more than one.

If put ?2=1, ?1=0, ?0=-1 in this system, then by solving the received system of nonlinear algebraic equations, we have:

Hence we get the following method:

     (24)

Remark, that this method is symmetric (so that v0=-v2 ). But there is the nun symmetric  method  with the degree p=9. 

It is clear, that for using the method (24), it is necessary to determine the values ??of the quantities yn+yo and yn+y2. To illustrate the above mentioned, consider the case k=1 and put ?1=?2=0. Then by solving the system (23), we obtain By using this solution in the formula (9), one receive  the method (10), for using which can be suggested the following algorithm, if is known the value y1/2.

Note that for solving some problems by this algorithm for calculating values ??of the quantity yi+3/2 can be used the following method:

To illustrate applying  this algorithm to solving problem (1)-(2) consider the following examples:

For solving these problems, we are using above mentioned algorithm. Note that the example 1 is solved in the work [2], the examples 1-3 are solved in the work [7], the example 3 is solved in the work [3], and the example 4 is solved in the work [3]. Here all the examples solved by the hybrid method (10) and the receive results are putting in table 1 and also to solving some of these  problems, here used the trapezoid method and the receive results are putting in the table 2,  in which we used the next notation:

Method I - Predictor-corrector method consist is Euler and Trapezoid method applying to solving system consists only of ODE.

Method II - The same predictor-corrector method applying to solving system consists only of the integral equations.

Method III - The same predictor-corrector method applying to solving system consists of  ODE and integral equation.  

Table 1.

Number of

the

examples

Value

of the

variable

Maximal

error

for the

method

from

[2]

Maximal

error

for the

first

method

from

[7]

Maximal

error

for the

second

method

from

[7]

Maximal

error

for the

method

from

[3]

Maximal

error

for the

hybrid

method

(10)

I h=0.05

II h=1/32

III h=1/8

IV h=1/8

1.0

3.0

2.0

2.0

0.12 E-05

0.16 E-04

 

0.86 E-05

0.43 E-03

0.77 E-06

 

0.16 E-05

0.58 E-04

 

0.18 E-06

0.2 E-05

0.129 E-03

0.11 E-05

0.21 E-04

0.52 E-06

0.2 E-04

 Table 2.

 

 

 

 

Method I

Method II

Method III

I h=0.05

II h=1/32

III h=0.05

IV h=1/32

1.0

2.0

1.0

2.0

0.28 E-03

0.12 E-03

0.19 E-03

0.68 E-03

0.24 E-03

0.33 E-03

0.19 E-03

0.58 E-03

0.52 E-04

0.70 E-04

0.19 E-03

0.58 E-03

Conclusion. By the above mentioned are shown some of the advantages of the hybrid methods. Constructed concrete hybrid methods with the high accuracy. And also, in the simple case, have constructed an algorithm for using to solving of the problems of type (1)-(2). Note that the proposed algorithm for the using of the method (10) has a simple structure, which makes easy to using and to modifying it. Naturally, each method has some advantages and shortcomings. The main advantage of hybrid methods is their high accuracy, and the main shortcomings is the using of variables with the values yn+y ??in irrational points  To overcome these shortcomings, here are using the methods of predictor-corrector type. It is easy to understand that the proposed algorithm can be modified by using more precise methods. We here describe a scheme by which to ensure applying of the hybrid method to solving problem (1). We believe that the study of hybrid methods is one of the promising directions, for the construction most accurate numerical methods, which also confirms by the above solved problems.

References:

  1. V.Volterra. Theory of functional and of integral and integro-differensial equations, (in Russian), Moscow, Nauka, 1982, 304 p.
  2. P.Linz Linear Multistep methods for Volterra Integro-Differensial equations, Journal of the Association for Computing Machinery, Vol.16, No.2, April 1969, pp. 295-301.
  3. Feldstein A, Sopka J.R. Numerical methods for nonlinear Volterra integro-differensial equations, SINUM 11, 826-846.
  4. H.Brunner. Imlicit Runge-Kutta Methods of Optimal oreder for Volterra integro-differensial equation. Methematics of computation, Volume 42, Number 165, January 1984, pp. 95-109.
  5. Makroglou A.A. Block – by-block method for the numerical solution of Volterra delay integro-differensial equations, Computing 3, 1983, 30, №1, p. 49-62.
  6. Bulatov M.B. Chistakov E.B. Chislennoe resheniye inteqro-differensialnix sisitem s virojdennoy matrisey pered proizvodnoy mnoqoshaqovimi metodami. Dif. Equations, 2006, 42, №9, pp. 1218-1255.
  7. A.Makroglou. Hybrid methods in the numerical solution of Volterra integro-differensial equations. Journal of Numerical Analysis 2, 1982, pp. 21-35.
  8. G.Y.Mehdiyeva, Ibrahimov V.R., M.N.Imanova. Application of the forward jumping Method to the solving of Volterra integral equation. Conference in Numerical Analysis, Chania, Crete, Greece. 2010, p. 106-111.
  9. G.Y.Mehdiyeva, Ibrahimov V.R., M.N.Imanova  On one application of hybrid methods for solving Volterre Integro-differential equations. World Academy of Science, engineering and Technology, Dubai, 2012, 1197-1201.
  10. G.Y.Mehdiyeva, Ibrahimov V.R., M.N.Imanova, R.R.Mirzayev.  On a way for constructing hybrid methods. Space and time – coordinate system of human development. Materials digest of VIIIth International Research and Practice Conference (Kiev, London, August 25 – September 1, 2011), p. 91-95.
  11. J.O.Ehigie, S.A.Okunuga, A.B.Sofoluwe, M.A.Akanbi. On generalized 2-step continuous linear multistep method of  hybrid type for the integration of second order ordinary differential equations, Archives of Applied Research, 2010, 2(6), pp.362-372.
  12. G.Y.Mehdiyeva, Ibrahimov V.R., Imanova M.N. Application of multi step methods to the solving second order ordinary differential equation. The Third international Conference “Problems of Cybernetics and Informatics”, Baku, Azerbaijan, 2010, p. 255-258.
  13. I.S.Berezin, N.P.Jidkov. Methodi vichisleniy (russian). Т.1, М. «Nauka», 1966, 632 с.
  14. Hamming R.W. Numerical methods for scientists and engineers. McGraw-Hill, New York, 1962.
  15. Ibrahimov V.R. On some properties of the Richardson extrapolation. Dif.eq. № 12, 1990, 2170-2173.
Комментарии: 1

Артамонова Елена Николаевна

Various applications of integro-differential equations, such as nuclear reactors, viscoelasticity, wave propagation and engineering systems. Restrictions on the use of Laplace transforms (the Laplace transform is related to the Fourier transform, but whereas the Fourier transform expresses a function or signal as a series of modes of vibration , the Laplace transform resolves a function into its moments)?
Комментарии: 1

Артамонова Елена Николаевна

Various applications of integro-differential equations, such as nuclear reactors, viscoelasticity, wave propagation and engineering systems. Restrictions on the use of Laplace transforms (the Laplace transform is related to the Fourier transform, but whereas the Fourier transform expresses a function or signal as a series of modes of vibration , the Laplace transform resolves a function into its moments)?
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